Problem: Captain William has a ship, the H.M.S Crimson Lynx. The ship is five furlongs from the dread pirate Emily and her merciless band of thieves. If his ship hasn't already been hit, Captain William has probability $\dfrac{1}{3}$ of hitting the pirate ship. If his ship has been hit, Captain William will always miss. If her ship hasn't already been hit, dread pirate Emily has probability $\dfrac{1}{7}$ of hitting the Captain's ship. If her ship has been hit, dread pirate Emily will always miss. If the Captain and the pirate each shoot once, and the Captain shoots first, what is the probability that the Captain hits the pirate ship, but the pirate misses?
The probability of event A happening, then event B, is the probability of event A happening times the probability of event B happening given that event A already happened. In this case, event A is the Captain hitting the pirate ship and event B is the pirate missing the Captain's ship. The Captain fires first, so his ship can't be sunk before he fires his cannons. So, the probability of the Captain hitting the pirate ship is $\dfrac{1}{3}$. If the Captain hit the pirate ship, the pirate has no chance of firing back. So, the probability of the pirate missing the Captain's ship given the Captain hitting the pirate ship is $1$. The probability that the Captain hits the pirate ship, but the pirate misses is then the probability of the Captain hitting the pirate ship times the probability of the pirate missing the Captain's ship given the Captain hitting the pirate ship. This is $\dfrac{1}{3} \cdot 1 = \dfrac{1}{3}$